In mathematics, an affine bundle is a fiber bundle whose typical fiber, fibers, trivialization morphisms and transition functions are affine.^[1]

Formal definition

Let ${\displaystyle {\overline {\pi )):{\overline {Y))\to X}$ be a vector bundle with a typical fiber a vector space ${\displaystyle {\overline {F))}$. An affine bundle modelled on a vector bundle ${\displaystyle {\overline {\pi )):{\overline {Y))\to X}$ is a fiber bundle ${\displaystyle \pi :Y\to X}$ whose typical fiber ${\displaystyle F}$ is an affine space modelled on ${\displaystyle {\overline {F))}$ so that the following conditions hold:

(i) Every fiber ${\displaystyle Y_{x))$ of ${\displaystyle Y}$ is an affine space modelled over the corresponding fibers ${\displaystyle {\overline {Y))_{x))$ of a vector bundle ${\displaystyle {\overline {Y))}$.

(ii) There is an affine bundle atlas of ${\displaystyle Y\to X}$ whose local trivializations morphisms and transition functions are affine isomorphisms.

Dealing with affine bundles, one uses only affine bundle coordinates ${\displaystyle (x^{\mu },y^{i})}$ possessing affine transition functions

${\displaystyle y'^{i}=A_{j}^{i}(x^{\nu })y^{j}+b^{i}(x^{\nu }).}$

There are the bundle morphisms

${\displaystyle Y\times _{X}{\overline {Y))\longrightarrow Y,\qquad (y^{i},{\overline {y))^{i})\longmapsto y^{i}+{\overline {y))^{i},}$

${\displaystyle Y\times _{X}Y\longrightarrow {\overline {Y)),\qquad (y^{i},y'^{i})\longmapsto y^{i}-y'^{i},}$

where ${\displaystyle ({\overline {y))^{i})}$ are linear bundle coordinates on a vector bundle ${\displaystyle {\overline {Y))}$, possessing linear transition functions ${\displaystyle {\overline {y))'^{i}=A_{j}^{i}(x^{\nu }){\overline {y))^{j))$.

Properties

An affine bundle has a global section, but in contrast with vector bundles, there is no canonical global section of an affine bundle. Let ${\displaystyle \pi :Y\to X}$ be an affine bundle modelled on a vector bundle ${\displaystyle {\overline {\pi )):{\overline {Y))\to X}$. Every global section ${\displaystyle s}$ of an affine bundle ${\displaystyle Y\to X}$ yields the bundle morphisms

${\displaystyle Y\ni y\to y-s(\pi (y))\in {\overline {Y)),\qquad {\overline {Y))\ni {\overline {y))\to s(\pi (y))+{\overline {y))\in Y.}$

In particular, every vector bundle ${\displaystyle Y}$ has a natural structure of an affine bundle due to these morphisms where ${\displaystyle s=0}$ is the canonical zero-valued section of ${\displaystyle Y}$. For instance, the tangent bundle ${\displaystyle TX}$ of a manifold ${\displaystyle X}$ naturally is an affine bundle.

An affine bundle ${\displaystyle Y\to X}$ is a fiber bundle with a general affine structure group ${\displaystyle GA(m,\mathbb {R} )}$ of affine transformations of its typical fiber ${\displaystyle V}$ of dimension ${\displaystyle m}$. This structure group always is reducible to a general linear group ${\displaystyle GL(m,\mathbb {R} )}$, i.e., an affine bundle admits an atlas with linear transition functions.

By a morphism of affine bundles is meant a bundle morphism ${\displaystyle \Phi :Y\to Y'}$ whose restriction to each fiber of ${\displaystyle Y}$ is an affine map. Every affine bundle morphism ${\displaystyle \Phi :Y\to Y'}$ of an affine bundle ${\displaystyle Y}$ modelled on a vector bundle ${\displaystyle {\overline {Y))}$ to an affine bundle ${\displaystyle Y'}$ modelled on a vector bundle ${\displaystyle {\overline {Y))'}$ yields a unique linear bundle morphism

${\displaystyle {\overline {\Phi )):{\overline {Y))\to {\overline {Y))',\qquad {\overline {y))'^{i}={\frac {\partial \Phi ^{i)){\partial y^{j))}{\overline {y))^{j},}$

called the linear derivative of ${\displaystyle \Phi }$.